Algebraic Criteria on Controllability and Positive-Controllability of Discrete-Time Bilinear Systems: Seeking Nonzero Entries

In this article, we consider controllability of discrete-time bilinear systems. A classical sufficient condition was obtained in this journal in 1973, which, unfortunately, is not algebraically verifiable, and the required control inputs to achieve state transition are unknown. However, based on a recently established notion, near-controllability, we propose a seeking nonzero entries approach to derive fully algebraically verifiable controllability criteria for discrete-time bilinear systems. More specifically, by our approach, the verification of controllability of the systems is transformed into the problem of seeking nonzero entries of a sequence of matrices generated by the two system matrices. In addition, the control inputs steering the controllable systems between any given pair of states can be easily computed by proposing two algorithms based on near-controllability. As applications, the seeking nonzero entries approach is generalized to construct arbitrary-finite-dimensional controllable discrete-time bilinear systems and also to prove positive-controllability of discrete-time bilinear systems, where algebraic criteria are obtained by the root locus theory and positive-near-controllability that is a new notion introduced in this article. Examples are provided to illustrate the obtained algebraic controllability criteria and algorithms.